Texture assigned to more than one object renders differently
The object which has another appearence of the texture was scaled in Object mode, while its UV map was left as it was from the original one. Apply the transforms, redo (or change) the UV map and the texture will act in the same way on both objects.

What happens when duplicating unwrapped object

The original object was unwrapped.
Then the duplicate was created. It derived the UV map from the original object, and that UV map was the same as there.
After that the duplicate was scaled to fit the scene specifics.

The difference consists in that the duplicate was scaled in Object mode, and most likely was scaled non-uniformly (e.g. by Z axis more than by X). These transforms weren't applied to the object data. Hence the mesh was different from one unfolded in the UV map, and that's where the difference came from.

Generally, if you scale by one axis in Edit mode, you'll get non-unform texture distribution too. However you will be able to change that by unwrapping once more.

For details about the possible cases when you need applying transforms refer to Why is it important to apply transformation to an objects data?